Efficient inference in multivariate fractionally integrated time series models. (English) Zbl 1094.91054
The author proposes maximum likelihood estimators for multivariate fractionally integrated time series models and develops a number of tests based on these estimators. The results generalise the previous work on univariate models. One of the main features of the author’s approach is that it leads to inference based on normal and \(\chi^2\)-distributions which is in striking contrast with the related work on unit roots and nonstationary models.
More specifically, maximum likelihood estimators based on the Gaussian likelihood are developed. These are used to derive Lagrange multipliers, likelihood ratio (LR), and Wald test statistics under local alternatives. The cases of fractional integration of the same order and different orders are considered separately. The error process may be white noise or a multivariate autoregression. The distributional results do not require normality but when that assumption is made the tests become asymptotically efficient.
The results may be used to test jointly various hypotheses about the integration orders of the components of the vector process and to construct joint confidence regions for them. Especially valuable is the possibility to test for common integration order without specifying a particular value for it. Extensive simulation evaluation of the finite sample performance of the LR statistic is presented.
More specifically, maximum likelihood estimators based on the Gaussian likelihood are developed. These are used to derive Lagrange multipliers, likelihood ratio (LR), and Wald test statistics under local alternatives. The cases of fractional integration of the same order and different orders are considered separately. The error process may be white noise or a multivariate autoregression. The distributional results do not require normality but when that assumption is made the tests become asymptotically efficient.
The results may be used to test jointly various hypotheses about the integration orders of the components of the vector process and to construct joint confidence regions for them. Especially valuable is the possibility to test for common integration order without specifying a particular value for it. Extensive simulation evaluation of the finite sample performance of the LR statistic is presented.
Reviewer: Georgi Boshnakov (Manchester)
Keywords:
multivariate ARFIMA model; local alternative; multivariate fractional unit root test; LR test; finite sample performanceReferences:
| [1] | Agiakloglou C., Journal of Time Series Analysis 15 pp 253– (1994) |
| [2] | Beran J., Journal of the Royal Statistical Society Series B 57 pp 659– (1995) |
| [3] | DOI: 10.1016/S0304-4076(02)00091-X · Zbl 1038.62075 · doi:10.1016/S0304-4076(02)00091-X |
| [4] | DOI: 10.1016/S0304-4076(98)00021-9 · Zbl 1070.62522 · doi:10.1016/S0304-4076(98)00021-9 |
| [5] | DOI: 10.1111/1468-0262.00359 · Zbl 1132.91590 · doi:10.1111/1468-0262.00359 |
| [6] | Doornik J. A., Ox: An Object-Oriented Matrix Language, 4. ed. (2001) |
| [7] | J. A. Doornik, and M. Ooms (2001 ). A package for estimating, forecasting and simulating arfima models : Arfima package 1.01 for Ox. Working Paper, Nuffield College, Oxford. |
| [8] | Fountis N. G., Annals of Statistics 17 pp 419– (1989) |
| [9] | Hall P., Martingale Limit Theory and its Application (1980) · Zbl 0462.60045 |
| [10] | Hosking J. R. M., Journal of the American Statistical Association 75 pp 602– (1980) |
| [11] | DOI: 10.1016/0304-4076(95)01738-0 · Zbl 0854.62085 · doi:10.1016/0304-4076(95)01738-0 |
| [12] | DOI: 10.1017/S0266466699154057 · Zbl 0985.62070 · doi:10.1017/S0266466699154057 |
| [13] | DOI: 10.1016/0165-1889(88)90041-3 · Zbl 0647.62102 · doi:10.1016/0165-1889(88)90041-3 |
| [14] | Kugler P., Journal of Applied Econometrics 8 pp 163– (1993) |
| [15] | DOI: 10.1017/S0266466601174049 · Zbl 1018.62075 · doi:10.1017/S0266466601174049 |
| [16] | DOI: 10.1111/1467-9892.00043 · Zbl 0938.62103 · doi:10.1111/1467-9892.00043 |
| [17] | Magnus J. R., Matrix Differential Calculus with Applications in Statistics and Econometrics (1999) · Zbl 0912.15003 |
| [18] | DOI: 10.1016/S0378-3758(98)00245-6 · Zbl 0934.60071 · doi:10.1016/S0378-3758(98)00245-6 |
| [19] | DOI: 10.1016/S0304-4149(99)00088-5 · Zbl 1028.60030 · doi:10.1016/S0304-4149(99)00088-5 |
| [20] | M. O. Nielsen (2002 ). Multivariate Lagrange multiplier tests for fractional integration .Department of Economics Working Paper 2002-18,University of Aarhus. |
| [21] | Nielsen M. O., Journal of Business and Economic Statistics (2004) |
| [22] | DOI: 10.1017/S0266466604201050 · Zbl 1046.62091 · doi:10.1017/S0266466604201050 |
| [23] | Phillips P. C. B., Review of Economic Studies 53 pp 473– (1986) |
| [24] | DOI: 10.1016/0304-4076(91)90078-R · Zbl 0734.62070 · doi:10.1016/0304-4076(91)90078-R |
| [25] | Robinson P. M., Journal of the American Statistical Association 89 pp 1420– (1994) |
| [26] | Sargan J. D., Econometrica 51 pp 799– (1983) |
| [27] | DOI: 10.1017/S0266466699154045 · Zbl 0985.62073 · doi:10.1017/S0266466699154045 |
| [28] | Watson M. W., Handbook of Econometrics, Vol. IV pp 2843– (1994) |
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